Sodium Chloride and Rutile-Related Structure Systems
2.4. CHARACTERISTICS OF RUTILE STRUCTURES
Rutile has a tetragonal unit cell in which the Ti atoms are located at (0,0,0) and (0.5,0.5,0.5) and the oxygen atoms at (x,x,O), (1 - x, 1 - x,O), (0.5 +x,0.5 - x,O.5), and (0.5 - x, 0.5
+
x, 0.5), with x=
0.3. The rutile structure (Fig. 2.6a) can be conveniently illustrated as follows. The basic mosaic or unit is Ti06 octahedron and the octahedra share parallel edges and form chains if viewed along the c axis, and the chains are connected by sharing the vertices, but the orientations of the two chains are different: one is rotated around the chain axis [001] for 900 as shown in Fig. 2.6b. As illustrated, the oxygen anions form a close-packing hexagonal sheet with equal-sided triangle units. If the titanium cations are located in every second row at the centers of the triangles, and the next close-packing oxygen layer is stacked on the cation layer in a way equivalent to the C stacking position of the second layer being superimposed on the B position of the first layer, as shown in Fig.2.6c, the rutile structure is created. Based on this consideration, the distribution of titanium cations implies that the Ti06 octahedra change their connections from comer sharing to edge sharing. Titanium is a 3d transition metal and preferably forms an octahedral complex in the oxidation states I, II, III, and IV. In titanium dioxide the Ti06 octahedron is the basic unit (or mosaic) to build the 3-D network. The structural evolution should be based on this unit, and changing the connection between them is the way for creating some new structures. There are several ways to connect the octahedra, as given below.2.4.1. APEX SHARING
Figure 2.7 shows three possible situations for apex sharing (or comer sharing): two 1800 connections, and one 131.80 inclined connection of two octahedra. The 1800 connection has the maximum separation between the adjacent cations. The interaction between the adjacent cations is a minimum. M-O-M or O-M-O is the basic unit to consider the electron band structure. In general, this is the lowest energy configuration.
The inclined 131.80 connection has the shorter distance between the adjacent cations, and the interaction between the adjacent cations is increased, resulting in higher energy. Of course, the inclined angle can vary from 131.80 to 1800 , and the connective apices also can vary from an opposite one to another as shown in Fig. 2.7d. It can occur in chain-like or ring-like structures.
From the chemical composition point of view, if the octahedra share opposite linear (180°) vertices the composition of the chain should be (MOs)n as shown in Fig. 2.8, where n stands for the number of octahedra in the chain. If four chains are connected with each other by sharing the remaining vertices, it must form a (M03)n 3-D network. If we use the basic unit to express the composition, this is an M03 compound and the structure is cubic. Using a different number of octahedron chains can create many different types
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of structures and compositions via different ways of sharing vertices. Figure 2.9 gives the M03 structure and its derivatives created by rotating the octahedron in different ways.
From the close-packing point of view the variation in octahedral connections implies distortion and corrugation. Figure 2.lOa gives the different polygons of the octahedron chains in two dimensions. The triangle, pentagonal, and hexagonal rings are usually seen in the structures of oxides. The hexagonal rings can fill two-dimensional space as shown in Fig. 2.l0b. The standard pentagonal rings can combine to form distorted triangles and!
or parallelograms, but they cannot fill two-dimensional space via translation. The
(b) Apex
Figure 2.6. (a) 3-D unit cell of rutile, in which the two types of octahedra rotated for 900 with each other are indicated. The anions located at the apexes are highlighted. (b) The [001] projection of the unit cell showing the arrangement of the octahedron chains along the c-axis direction. (c) Ideal rutile structure showing the oxygen close-packing layers and the two types of chains formed by edge-sharing octahedra. The chains are connected via comer sharing.
(a) 1800 (b) 1800
Figure 2.7. Configurations of two comer-sharing octahedra. There are (a,b) two 1800 and (c) one 131.80 connection in which the common comer and its nearest three neighbors have a tetrahedron array (indicated by dashed lines). (d) A case between 1800 and 131.80
(MOS)n comer- haring chains with different connective angle
Figure 2.8. The comer-sharing connections among octahedra. The one-dimensional comer-sharing octahedra have the composition (MOs)m where n is the number of octahedra in the chain. The ring of the comer-sharing octahedra has a composition of M03 .
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pentagonal rings with some distortions may fill two-dimensional space (Fig. 2.10c). The simplest four octahedra rings of M03 are usually called Re03 structures.
2.4.2. EDGE SHARING
Figure 2.11a shows several possible octahedron edge-sharing structures. The distance between the adjacent cations is shorter than that in the one with 1800 vertex sharing and less screening because the cation is at the middle point of two adjacent
Figure 2.9. (a) The 3-D corner-sharing octahedral arrays of the M03 structure and its [001] and [111] views.
The structure is formed by corner-sharing octahedron chains linked also by corner sharing. (b) The formation of other structures by bending and rotating the corner-sharing octahedra chains but still keep corner sharing between the chains.
anions. Therefore, the interaction between the positively charged cations is stronger.
Regarding the valence state of the cations it is possible and easier to have displacement of the cations from the centers of the octahedra as shown in Fig. 2.11 b, resulting in different lengths of the M-O bonding. This variation will change the band structure and the property ofthe compound due to the Peierls distortion effect (Section 1.10.1). In V02, for example, the vanadium cations have been displaced from the centers of the octahedra, inducing a structure transformation from rutile to monoclinic. Therefore the electron band structure also has been changed: band broadening due to shorter and longer M-O bonding distances, resulting in property changes from semiconductive in rutile structure to metallic in monoclinic (Goodenough, 1971).
Sharing two opposite edges has higher energy, and it is rare to share all of the edges of the octahedra except in the cases where the cation and anion have appropriate radius ratio (rM/rx > 0.414) and pure ionic bonds as in sodium chloride.
Figure 2.10. Linear comer-sharing octahedron chains connected by sharing the remaining comers. (a) Rings formed by three, five, and six linear octahedron chains. (b) The rings of three and six chains can be combined to fill the 2-D space, (c) but the ring of five chains cannot fill the space unless they are distorted.
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(a)
(b)
Figure 2.11. Arrays of edge-sharing octahedra. (a) The linear, zigzag, ring, and sheet of octahedra formed by edge-sharing. (b) The linear edge-sharing octahedra with cation distortion because of strong interaction.
2.4.3. FACE SHARING
The face-sharing octahedra hardly occur in rutile-related structures because the adjacent cations are too close to each other and the screening effect from the anions is small. The interaction energy is too high. However, NaCl and CsCI have the face-sharing octahedral arrays due to the radius ratio (rM/rX > 0.414) and pure ionic bonding.